Farey Fractions and Ford Circles

The most convincing derivation of periodic structure, using the concepts of number theory, comes from a comparison with Farey sequences. The Farey scheme is a device to arrange rational fractions in enumerable order. Starting from the end members of the interval [0,1] an infinite tree structure is generated by separate addition of numerators and denominators to produce the Farey sequences ni of order n, where n limits the values of denominators 

A pair of fractions, adjacent to each other in any sequence, has the property of unimodularity, such that for the pair h/k, l/m the quantity hm — kl = 1. 

Each rational fraction, h/k, defines a Ford circle with a radius and y-coordinate of 1/(2k2), positioned at an -coordinate h/k. The Ford circles of any unimodular pair are tangent to each other and to the x-axis. The circles, numbered from 1 to 4 in the construction overleaf, represent the Farey sequence of order 4. This sequence has the remarkable property of one-to-one correspondence with the natural numbers ordered in sets of 2k2 and in the same geometrical relationship as the Ford circles of 4. 

The detail of secondary periods embedded within primaries, e.g. 2 within 8, 8 within 18, etc., to generate substructures such as 2 + 6 = 8, 8 +10 = 18, 

18 + 14 = 32, is correctly mapped by the Ford circles that represent nonprimitive fractions. A nesting of Ford circles 1/1, 2/2, 3/3, 4/4. .., is shown by way of illustration. 

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